UniformIsingModels

A fully-connected ferromagnetic Ising model with uniform coupling strength, described by a Boltzmann distribution

$p(\boldsymbol{\sigma}) = \frac{1}{Z} \exp\left[\beta\left(\frac{J}{N}\sum_{i<j}\sigma_i\sigma_j+\sum_{i=1}^Nh_i\sigma_i\right)\right],\quad \boldsymbol{\sigma}\in\{-1,1\}^N $

is exactly solvable in polynomial time.

Quantity	Expression	Cost
Normalization	$Z=\sum\limits_{\boldsymbol{\sigma}}\exp\left[\beta\left(\frac{J}{N}\sum_{i<j}\sigma_i\sigma_j+\sum_{i=1}^Nh_i\sigma_i\right)\right]$	$\mathcal O (N^2)$
Free energy	$F = -\frac{1}{\beta}\log Z$	$\mathcal O (N^2)$
Sample a configuration	$\boldsymbol{\sigma} \sim p(\boldsymbol{\sigma})$	$\mathcal O (N^2)$
Average energy	$U = \sum\limits_{\boldsymbol{\sigma}}p(\boldsymbol{\sigma})\left[-\left(\frac{J}{N}\sum_{i<j}\sigma_i\sigma_j+\sum_{i=1}^Nh_i\sigma_i\right)\right]$	$\mathcal O (N^2)$
Entropy	$S = -\sum\limits_{\boldsymbol{\sigma}}p(\boldsymbol{\sigma})\log p(\boldsymbol{\sigma})$	$\mathcal O (N^2)$
Distribution of the sum of the N spins	$p_S(s)=\sum\limits_{\boldsymbol{\sigma}}p(\boldsymbol{\sigma})\delta\left(s-\sum_{i=1}^N\sigma_i\right)$	$\mathcal O (N^2)$
Site magnetizations	$m_i=\sum\limits_{\boldsymbol{\sigma}}p(\boldsymbol{\sigma})\sigma_i,\quad\forall i\in{1,2,\ldots,N}$	$\mathcal O (N^3)$
Correlations	$r_{ij}=\sum\limits_{\boldsymbol{\sigma}}p(\boldsymbol{\sigma})\sigma_i\sigma_j,\quad\forall j\in{1,2,\ldots,N},i<j$	$\mathcal O (N^5)$

Example

]add UniformIsingModels

Construct a UniformIsing instance

using UniformIsingModels, Random

N = 10
J = 2.0
rng = MersenneTwister(0)
h = randn(rng, N)
β = 0.1
x = UniformIsing(N, J, h, β)

Compute stuff

# normalization and free energy
Z = normalization(x)
F = free_energy(x)

# energy and probability of a configuration
σ = rand(rng, (-1,1), N) 
E = energy(x, σ)
prob = pdf(x, σ)

# a sample along with its probability 
σ, p = sample(rng, x)

# single-site magnetizations <σᵢ>
m = site_magnetizations(x)

# distribution of the sum Σᵢσᵢ of all variables
ps = sum_distribution(x)

# energy expected value
U = avg_energy(x)

# entropy
S = entropy(x)

# correlations <σᵢσⱼ> and covariances <σᵢσⱼ>-<σᵢ><σⱼ>
p = correlations(x)
c = covariances(x)

Notes

The internals rely on dynamic programming.

If you know of any implementation that's more efficient than this one I'd be very happy to learn about it!